Population systems with nonlinear discrete diffusion terms.
type: Item
instance of: scholarly article
title: Population systems with nonlinear discrete diffusion terms.
zbMATH Open document ID: 1037.34040
DOI: 10.1016/S0022-247X(03)00249-X
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: The authors investigate the asymptotic behavior of the period population system with nonlinear discrete diffusion terms \[ \begin{aligned} x_i'&= D_i(t,x_1,\ldots,x_n)+ x_if_i(t,x_i,y_i),\\ y_i'&= E_i(t,y_1,\ldots,y_n)+y_ig_i(t,x_i,y_i), \end{aligned}\quad 1\leq i\leq n, \] where \(f_i,g_i: \mathbb{R}\times \mathbb{R}_+^2\rightarrow \mathbb{R}\) are continuous functions which are \(T\)-periodic in \(t\) and locally Lipschitz continuous in \((x_i,y_i)\) and \(D=(D_1,\ldots,D_n),\) \(E=(E_1,\ldots,E_n)\) are \(T\)-periodic nonlinear diffusion functions. For a predator-prey system, the authors prove by using Brower's degree theory, that there exists a positive \(T\)-periodic solution under some conditions and generalize the extinction result of \textit{J. López-Gómez, R. Ortega} and the second author [Adv. Differ. Equ. 1, 403--423 (1996; Zbl 0849.34026)]. For a competitive system, they present by using the method of monotone iterative schemes conditions for the global asymptotic stability of trivial and semitrivial \(T\)-periodic solutions, further, conditions for the existence of an unstable coexistence state are also investigated.
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 1956894
zbMATH Keywords: population systems
zbMATH Keywords: Nonlinear diffusion
zbMATH Keywords: Predator-prey
zbMATH Keywords: competitive system
zbMATH Keywords: Brouwer's degree
zbMATH Keywords: Fixed-point index
zbMATH Keywords: Iterative schemes
author: M. Gámez
author: Antonio Tineo
reviewed by: Lansun Chen
MaRDI profile type: Publication
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full work available at URL: s0022-247x(03)00249-x
OpenAlex ID: W2012995251
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description: scientific article; zbMATH DE number 1956894